全面的品質(zhì)管控培訓(xùn)資料(英文版)課件

本資料來源本資料來源全面的品質(zhì)管控培訓(xùn)資料(英文版)課件Chapter17QualityControlChapter17QualityControlOverviewIntroductionStatisticalConceptsinQualityControlControlChartsAcceptancePlansComputersinQualityControlQualityControlinServicesWrap-Up:WhatWorld-ClassCompaniesDoOverviewIntroductionIntroductionQualitycontrol(QC)includestheactivitiesfromthesuppliers,throughproduction,andtothecustomers.Incomingmaterialsareexaminedtomakesuretheymeettheappropriatespecifications.Thequalityofpartiallycompletedproductsareanalyzedtodetermineifproductionprocessesarefunctioningproperly.Finishedgoodsandservicesarestudiedtodetermineiftheymeetcustomerexpectations.IntroductionQualitycontrol(QQCThroughoutProductionSystemsRawMaterials,Parts,andSuppliesProductionProcessesProductsandServicesInputsConversionOutputsControlChartsandAcceptanceTestsControlChartsandAcceptanceTestsControlChartsQualityofInputsQualityofOutputsQualityofPartiallyCompletedProductsQCThroughoutProductionSysteServicesandTheirCustomerExpectationsHospitalPatientreceivethecorrecttreatments?Patienttreatedcourteouslybyallpersonnel?Hospitalenvironmentsupportpatientrecovery?BankCustomer’stransactionscompletedwithprecision?Bankcomplywithgovernmentregulations?Customer’sstatementsaccurate?ServicesandTheirCustomerExProductsandTheirCustomerExpectationsAutomakerAutohavetheintendeddurability?Partswithinthemanufacturingtolerances?Auto’sappearancepleasing?LumbermillLumberwithinmoisturecontenttolerances?Lumberproperlygraded?Knotholes,splits,andotherdefectsexcessive?ProductsandTheirCustomerExSamplingTheflowofproductsisbrokenintodiscretebatchescalledlots.Randomsamplesareremovedfromtheselotsandmeasuredagainstcertainstandards.Arandomsampleisoneinwhicheachunitinthelothasanequalchanceofbeingincludedinthesample.Ifasampleisrandom,itislikelytoberepresentativeofthelot.SamplingTheflowofproductsiSamplingEitherattributesorvariablescanbemeasuredandcomparedtostandards.Attributesarecharacteristicsthatareclassifiedintooneoftwocategories,usuallydefective(notmeetingspecifications)ornondefective(meetingspecifications).Variablesarecharacteristicsthatcanbemeasuredonacontinuousscale(weight,length,etc.).SamplingEitherattributesorvSizeandFrequencyofSamplesAsthepercentageoflotsinsamplesisincreased:thesamplingandsamplingcostsincrease,andthequalityofproductsgoingtocustomersincreases.Typically,verylargesamplesaretoocostly.Extremelysmallsamplesmightsufferfromstatisticalimprecision.Largersamplesareordinarilyusedwhensamplingforattributesthanforvariables.SizeandFrequencyofSamplesAWhentoInspect

DuringtheProductionProcessInspectbeforecostlyoperations.Inspectbeforeoperationsthatarelikelytoproducefaultyitems.Inspectbeforeoperationsthatcoverupdefects.Inspectbeforeassemblyoperationsthatcannotbeundone.Onautomaticmachines,inspectfirstandlastpiecesofproductionruns,butfewin-betweenpieces.Inspectfinishedproducts.WhentoInspect

DuringtheProCentralLimitTheoremThecentrallimittheoremis:Samplingdistributionscanbeassumedtobenormallydistributedeventhoughthepopulation(lot)distributionsarenotnormal.Thetheoremallowsuseofthenormaldistributiontoeasilysetlimitsforcontrolchartsandacceptanceplansforbothattributesandvariables.CentralLimitTheoremThecentrSamplingDistributionsThesamplingdistributioncanbeassumedtobenormallydistributedunlesssamplesize(n)isextremelysmall.Themeanofthesamplingdistribution(x)isequaltothepopulationmean(m).Thestandarderrorofthesamplingdistribution(sx)issmallerthanthepopulationstandarddeviation(sx)byafactorof1/=-SamplingDistributionsThesampPopulationandSamplingDistributionsf(x)PopulationDistributionSamplingDistributionofSampleMeansMean=mStd.Dev.=sxMean=x=mStd.Error==xPopulationandSamplingDistriControlChartsPrimarypurposeofcontrolchartsistoindicateataglancewhenproductionprocessesmighthavechangedsufficientlytoaffectproductquality.Iftheindicationisthatproductqualityhasdeteriorated,orislikelyto,thencorrectiveistaken.Iftheindicationisthatproductqualityisbetterthanexpected,thenitisimportanttofindoutwhysothatitcanbemaintained.Useofcontrolchartsisoftenreferredtoasstatisticalprocesscontrol(SPC).ControlChartsPrimarypurposeConstructingControlChartsVerticalaxisprovidesthescaleforthesampleinformationthatisplottedonthechart.Horizontalaxisisthetimescale.Horizontalcenterlineisideallydeterminedfromobservingthecapabilityoftheprocess.Twoadditionalhorizontallines,theloweranduppercontrollimits,typicallyare3standarddeviationsbelowandabove,respectively,thecenterline.ConstructingControlChartsVerConstructingControlChartsIfthesampleinformationfallswithintheloweranduppercontrollimits,thequalityofthepopulationisconsideredtobeincontrol;otherwisequalityisjudgedtobeoutofcontrolandcorrectiveactionshouldbeconsidered.TwoversionsofcontrolchartswillbeexaminedControlchartsforattributesControlchartsforvariablesConstructingControlChartsIfControlChartsforAttributesInspectionoftheunitsinthesampleisperformedonanattribute(defective/non-defective)basis.Informationprovidedfrominspectingasampleofsizenisthepercentdefectiveinasample,p,orthenumberofunitsfoundtobedefectiveinthatsampledividedbyn.ControlChartsforAttributesAlthoughthedistributionofsampleinformationfollowsabinomialdistribution,thatdistributioncanbeapproximatedbyanormaldistributionwithameanofpstandarddeviationofThe3scontrollimitsareControlChartsforAttributes-AlthoughthedistributionofsExample:AttributeControlChart

EverycheckcashedordepositedatLincolnBankmustbeencodedwiththeamountofthecheckbeforeitcanbegintheFederalReserveclearingprocess.Theaccuracyofthecheckencodingprocessisofupmostimportance.Ifthereisanydiscrepancybetweentheamountacheckismadeoutforandtheencodedamount,thecheckisdefective.Example:AttributeControlChExample:AttributeControlChart

Twentysamples,eachconsistingof250checks,wereselectedandexamined.Thenumberofdefectivechecksfoundineachsampleisshownbelow.Example:AttributeControlChExample:AttributeControlChart

Themanagerofthecheckencodingdepartmentknowsfrompastexperiencethatwhentheencodingprocessisincontrol,anaverageof1.6%oftheencodedchecksaredefective. Shewantstoconstructapchartwith3-standarddeviationcontrollimits.Example:AttributeControlChExample:AttributeControlChartExample:AttributeControlChExample:AttributeControlChartExample:AttributeControlChInspectionoftheunitsinthesampleisperformedonavariablebasis.Theinformationprovidedfrominspectingasampleofsizenis:Samplemean,x,orthesumofmeasurementofeachunitinthesampledividedbynRange,R,ofmeasurementswithinthesample,orthehighestmeasurementinthesampleminusthelowestmeasurementinthesampleControlChartsforVariablesInspectionoftheunitsintheInthiscasetwoseparatecontrolchartsareusedtomonitortwodifferentaspectsoftheprocess’soutput:CentraltendencyVariabilityCentraltendencyoftheoutputismonitoredusingthex-chart.VariabilityoftheoutputismonitoredusingtheR-chart.ControlChartsforVariablesInthiscasetwoseparatecontx-ChartThecentrallineisx,thesumofanumberofsamplemeanscollectedwhiletheprocesswasconsideredtobe“incontrol”dividedbythenumberofsamples.The3slowercontrollimitisx-ARThe3suppercontrollimitisx+ARFactorAisbasedonsamplesize.===x-ChartThecentrallineisx,R-ChartThecentrallineisR,thesumofanumberofsamplerangescollectedwhiletheprocesswasconsideredtobe“incontrol”dividedbythenumberofsamples.The3slowercontrollimitisD1R.The3suppercontrollimitisD2R.FactorsD1andD2arebasedonsamplesize.R-ChartThecentrallineisR,3sControlChartFactorsforVariables

ControlLimitFactor ControlLimitFactor Sample forSampleMean forSampleRange Sizen A D1 D2 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.116 10 0.308 0.223 1.777 15 0.223 0.348 1.652 20 0.180 0.414 1.586 25 0.153 0.459 1.541 Over25 0.45+.001n 1.55-.0015n3sControlChartFactorsforVExample:VariableControlChart

HarryCoateswantstoconstructxandRchartsatthebag-fillingoperationforMeowChowcatfood.Hehasdeterminedthatwhenthefillingoperationisfunctioningcorrectly,bagsofcatfoodaverage50.01poundsandregularly-taken5-bagsampleshaveanaveragerangeof.322pounds.Example:VariableControlChaExample:VariableControlChartSampleMeanChart

x=50.01,R=.322,n=5 UCL=x+AR=50.01+.577(.322)=50.196 LCL=x

-AR

=50.01-.577(.322)=49.824===Example:VariableControlChaExample:VariableControlChartExample:VariableControlChaExample:VariableControlChartSampleRangeChart

x=50.01,R=.322,n=5 UCL=RD2=.322(2.116)=.681

LCL=RD1=.322(0)=0=Example:VariableControlChaExample:VariableControlChartExample:VariableControlChaAcceptancePlansTrendtodayistowarddevelopingtestingmethodsthataresoquick,effective,andinexpensivethatproductsaresubmittedto100%inspection/testingEveryproductshippedtocustomersisinspectedandtestedtodetermineifitmeetscustomerexpectationsButtherearesituationswherethisiseitherimpractical,impossibleoruneconomicalDestructivetests,wherenoproductssurvivetestInthesesituations,acceptanceplansaresensibleAcceptancePlansTrendtodayisAcceptancePlansAnacceptanceplanistheoverallschemeforeitheracceptingorrejectingalotbasedoninformationgainedfromsamples.Theacceptanceplanidentifiesthe:Sizeofsamples,nTypeofsamplesDecisioncriterion,c,usedtoeitheracceptorrejectthelotSamplesmaybeeithersingle,double,orsequential.AcceptancePlansAnacceptanceSingle-SamplingPlanAcceptanceorrejectiondecisionismadeafterdrawingonlyonesamplefromthelot.Ifthenumberofdefectives,c’,doesnotexceedtheacceptancecriteria,c,thelotisaccepted.Single-SamplingPlanAcceptanceSingle-SamplingPlanLotofNItemsRandomSampleofnItemsN-nItemsInspectnItemsc’>cc’<cReplaceDefectivesnNondefectivesc’DefectivesFoundinSampleRejectLotAcceptLotSingle-SamplingPlanLotofNIDouble-SamplingPlanOnesmallsampleisdrawninitially.Ifthenumberofdefectivesislessthanorequaltosomelowerlimit,thelotisaccepted.Ifthenumberofdefectivesisgreaterthansomeupperlimit,thelotisrejected.Ifthenumberofdefectivesisneither,asecondlargersampleisdrawn.Lotiseitheracceptedorrejectedonthebasisoftheinformationfrombothofthesamples.Double-SamplingPlanOnesmallDouble-SamplingPlanLotofNItemsRandomSampleofn1ItemsN–n1ItemsInspectn1Itemsc1’>c2c1’<c1ReplaceDefectivesn1Nondefectivesc1’DefectivesFoundinSampleRejectLotAcceptLotContinuec1

<c1’<c2(tonextslide)Double-SamplingPlanLotofNI(c1’+c2’)>c2Double-SamplingPlanN–n1ItemsRandomSampleofn2ItemsN–(n1+n2)ItemsInspectn2ItemsReplaceDefectivesn2Nondefectivesc2’DefectivesFoundinSampleRejectLotAcceptLotContinue(c1’+c2’)<c2(frompreviousslide)(c1’+c2’)>c2Double-SamplinSequential-SamplingPlanUnitsarerandomlyselectedfromthelotandtestedonebyone.Aftereachonehasbeentested,areject,accept,orcontinue-samplingdecisionismade.Samplingprocesscontinuesuntilthelotisacceptedorrejected.Sequential-SamplingPlanUnitsSequential-SamplingPlan

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312UnitsSampled(n)6NumberofDefectives4570RejectLotAcceptLotContinueSamplingSequential-SamplingPlan0DefinitionsAcceptanceplan-Samplesize(n)andmaximumnumberofdefectives(c)thatcanbefoundinasampletoacceptalotAcceptablequalitylevel(AQL)-IfalothasnomorethanAQLpercentdefectives,itisconsideredagoodlotLottolerancepercentdefective(LTPD)-IfalothasgreaterthanLTPD,itisconsideredabadlotDefinitionsAcceptanceplan-SDefinitionsAverageoutgoingquality(AOQ)–Giventheactual%ofdefectivesinlotsandaparticularsamplingplan,theAOQistheaverage%defectivesinlotsleavinganinspectionstationAverageoutgoingqualitylimit(AOQL)–Givenaparticularsamplingplan,theAOQListhemaximumAOQthatcanoccurastheactual%defectivesinlotsvariesDefinitionsAverageoutgoingquDefinitionsTypeIerror-Basedonsampleinformation,agood(quality)populationisrejectedTypeIIerror-Basedonsampleinformation,abad(quality)populationisacceptedProducer’srisk(a)-Foraparticularsamplingplan,theprobabilitythataTypeIerrorwillbecommittedConsumer’srisk(b)-Foraparticularsamplingplan,theprobabilitythataTypeIIerrorwillbecommittedDefinitionsTypeIerror-BaseConsiderationsin

SelectingaSamplingPlanOperatingcharacteristics(OC)curveAverageoutgoingquality(AOQ)curveConsiderationsin

SelectingaOperatingCharacteristic(OC)CurveAnOCcurveshowshowwellaparticularsamplingplan(n,c)discriminatesbetweengoodandbadlots.Theverticalaxisistheprobabilityofacceptingalotforaplan.Thehorizontalaxisistheactualpercentdefectiveinanincominglot.Foragivensamplingplan,pointsfortheOCcurvecanbedevelopedusingthePoissonprobabilitydistributionOperatingCharacteristic(OC)OperatingCharacteristic(OC)Curve.0.80.90ProbabilityofAcceptingtheLot

05101520251.00%DefectivesinLotsAQL=3%LTPD=15%Consumer’sRisk(b)=8.74%Producer’sRisk(a)=3.67%n=15,c=0OperatingCharacteristic(OC)OCCurve(continued)Managementmaywantto:Specifytheperformanceofthesamplingprocedurebyidentifyingtwopointsonthegraph:AQLanda

LTPDandbThenfindthecombinationofnandcthatprovidesacurvethatpassesthroughbothpointsOCCurve(continued)ManagementAverageOutgoingQuality(AOQ)CurveAOQcurveshowsinformationdepictedontheOCcurveinadifferentform.HorizontalaxisisthesameasthehorizontalaxisfortheOCcurve(percentdefectiveinalot).Verticalaxisistheaveragequalitythatwillleavethequalitycontrolprocedureforaparticularsamplingplan.Averagequalityiscalculatedbasedontheassumptionthatlotsthatarerejectedare100%inspectedbeforeenteringtheproductionsystem.AverageOutgoingQuality(AOQ)AOQCurveUnderthisassumption, AOQ=p[P(A)]/1 where:p=percentdefectiveinanincominglotP(A)=probabilityofacceptingalotisobtainedfromtheplan’sOCcurveAsthepercentdefectiveinalotincreases,AOQwillincreasetoapointandthendecrease.AOQCurveUnderthisassumptionAOQCurveAOQvaluewherethemaximumisattainedisreferredtoastheaverageoutgoingqualitylevel(AOQL).AOQListheworstaveragequalitythatwillexitthequalitycontrolprocedureusingthesamplingplannandc.AOQCurveAOQvaluewherethemComputersinQualityControlRecordsaboutqualitytestingandresultslimitafirm’sexposureintheeventofaproductliabilitysuit.RecallprogramsrequirethatmanufacturersKnowthelotnumberofthepartsthatareresponsibleforthepotentialdefectsHaveaninformationstoragesystemthatcantiethelotnumbersofthesuspectedpartstothefinalproductmodelnumbersHaveaninformationsystemthatcantrackthemodelnumbersoffinalproductstocustomersComputersinQualityControlReComputersinQualityControlWithautomation,inspectionandtestingcanbesoinexpensiveandquickthatcompaniesmaybeabletoincreasesamplesizesandthefrequencyofsamples,thusattainingmoreprecisioninbothcontrolchartsandacceptanceplansComputersinQualityControlWiQualityControlinServicesInallservicesthereisacontinuingneedtomonitorqualityControlchartsareusedextensivelyinservicestomonitorandcontroltheirqualitylevelsQualityControlinServicesInWrap-Up:World-ClassPracticeQualitycannotbeinspectedintoproducts.Processesmustbeoperatedtoachievequalityconformance;qualitycontrolisusedtoachievethis.Statisticalcontrolchartsareusedextensivelytoprovidefeedbacktoeveryoneaboutqualityperformance...moreWrap-Up:World-ClassPracticeQWrap-Up:World-ClassPracticeWhere100%inspectionandtestingareimpractical,uneconomical,orimpossible,acceptanceplansmaybeusedtodetermineiflotsofproductsarelikelytomeetcustomerexpectations.Thetrendistoward100%inspectionandtesting;automatedinspectionandtestinghasmadesuchanapproacheffectiveandeconomical.Wrap-Up:World-ClassPracticeW本資料來源本資料來源全面的品質(zhì)管控培訓(xùn)資料(英文版)課件Chapter17QualityControlChapter17QualityControlOverviewIntroductionStatisticalConceptsinQualityControlControlChartsAcceptancePlansComputersinQualityControlQualityControlinServicesWrap-Up:WhatWorld-ClassCompaniesDoOverviewIntroductionIntroductionQualitycontrol(QC)includestheactivitiesfromthesuppliers,throughproduction,andtothecustomers.Incomingmaterialsareexaminedtomakesuretheymeettheappropriatespecifications.Thequalityofpartiallycompletedproductsareanalyzedtodetermineifproductionprocessesarefunctioningproperly.Finishedgoodsandservicesarestudiedtodetermineiftheymeetcustomerexpectations.IntroductionQualitycontrol(QQCThroughoutProductionSystemsRawMaterials,Parts,andSuppliesProductionProcessesProductsandServicesInputsConversionOutputsControlChartsandAcceptanceTestsControlChartsandAcceptanceTestsControlChartsQualityofInputsQualityofOutputsQualityofPartiallyCompletedProductsQCThroughoutProductionSysteServicesandTheirCustomerExpectationsHospitalPatientreceivethecorrecttreatments?Patienttreatedcourteouslybyallpersonnel?Hospitalenvironmentsupportpatientrecovery?BankCustomer’stransactionscompletedwithprecision?Bankcomplywithgovernmentregulations?Customer’sstatementsaccurate?ServicesandTheirCustomerExProductsandTheirCustomerExpectationsAutomakerAutohavetheintendeddurability?Partswithinthemanufacturingtolerances?Auto’sappearancepleasing?LumbermillLumberwithinmoisturecontenttolerances?Lumberproperlygraded?Knotholes,splits,andotherdefectsexcessive?ProductsandTheirCustomerExSamplingTheflowofproductsisbrokenintodiscretebatchescalledlots.Randomsamplesareremovedfromtheselotsandmeasuredagainstcertainstandards.Arandomsampleisoneinwhicheachunitinthelothasanequalchanceofbeingincludedinthesample.Ifasampleisrandom,itislikelytoberepresentativeofthelot.SamplingTheflowofproductsiSamplingEitherattributesorvariablescanbemeasuredandcomparedtostandards.Attributesarecharacteristicsthatareclassifiedintooneoftwocategories,usuallydefective(notmeetingspecifications)ornondefective(meetingspecifications).Variablesarecharacteristicsthatcanbemeasuredonacontinuousscale(weight,length,etc.).SamplingEitherattributesorvSizeandFrequencyofSamplesAsthepercentageoflotsinsamplesisincreased:thesamplingandsamplingcostsincrease,andthequalityofproductsgoingtocustomersincreases.Typically,verylargesamplesaretoocostly.Extremelysmallsamplesmightsufferfromstatisticalimprecision.Largersamplesareordinarilyusedwhensamplingforattributesthanforvariables.SizeandFrequencyofSamplesAWhentoInspect

DuringtheProductionProcessInspectbeforecostlyoperations.Inspectbeforeoperationsthatarelikelytoproducefaultyitems.Inspectbeforeoperationsthatcoverupdefects.Inspectbeforeassemblyoperationsthatcannotbeundone.Onautomaticmachines,inspectfirstandlastpiecesofproductionruns,butfewin-betweenpieces.Inspectfinishedproducts.WhentoInspect

DuringtheProCentralLimitTheoremThecentrallimittheoremis:Samplingdistributionscanbeassumedtobenormallydistributedeventhoughthepopulation(lot)distributionsarenotnormal.Thetheoremallowsuseofthenormaldistributiontoeasilysetlimitsforcontrolchartsandacceptanceplansforbothattributesandvariables.CentralLimitTheoremThecentrSamplingDistributionsThesamplingdistributioncanbeassumedtobenormallydistributedunlesssamplesize(n)isextremelysmall.Themeanofthesamplingdistribution(x)isequaltothepopulationmean(m).Thestandarderrorofthesamplingdistribution(sx)issmallerthanthepopulationstandarddeviation(sx)byafactorof1/=-SamplingDistributionsThesampPopulationandSamplingDistributionsf(x)PopulationDistributionSamplingDistributionofSampleMeansMean=mStd.Dev.=sxMean=x=mStd.Error==xPopulationandSamplingDistriControlChartsPrimarypurposeofcontrolchartsistoindicateataglancewhenproductionprocessesmighthavechangedsufficientlytoaffectproductquality.Iftheindicationisthatproductqualityhasdeteriorated,orislikelyto,thencorrectiveistaken.Iftheindicationisthatproductqualityisbetterthanexpected,thenitisimportanttofindoutwhysothatitcanbemaintained.Useofcontrolchartsisoftenreferredtoasstatisticalprocesscontrol(SPC).ControlChartsPrimarypurposeConstructingControlChartsVerticalaxisprovidesthescaleforthesampleinformationthatisplottedonthechart.Horizontalaxisisthetimescale.Horizontalcenterlineisideallydeterminedfromobservingthecapabilityoftheprocess.Twoadditionalhorizontallines,theloweranduppercontrollimits,typicallyare3standarddeviationsbelowandabove,respectively,thecenterline.ConstructingControlChartsVerConstructingControlChartsIfthesampleinformationfallswithintheloweranduppercontrollimits,thequalityofthepopulationisconsideredtobeincontrol;otherwisequalityisjudgedtobeoutofcontrolandcorrectiveactionshouldbeconsidered.TwoversionsofcontrolchartswillbeexaminedControlchartsforattributesControlchartsforvariablesConstructingControlChartsIfControlChartsforAttributesInspectionoftheunitsinthesampleisperformedonanattribute(defective/non-defective)basis.Informationprovidedfrominspectingasampleofsizenisthepercentdefectiveinasample,p,orthenumberofunitsfoundtobedefectiveinthatsampledividedbyn.ControlChartsforAttributesAlthoughthedistributionofsampleinformationfollowsabinomialdistribution,thatdistributioncanbeapproximatedbyanormaldistributionwithameanofpstandarddeviationofThe3scontrollimitsareControlChartsforAttributes-AlthoughthedistributionofsExample:AttributeControlChart

EverycheckcashedordepositedatLincolnBankmustbeencodedwiththeamountofthecheckbeforeitcanbegintheFederalReserveclearingprocess.Theaccuracyofthecheckencodingprocessisofupmostimportance.Ifthereisanydiscrepancybetweentheamountacheckismadeoutforandtheencodedamount,thecheckisdefective.Example:AttributeControlChExample:AttributeControlChart

Twentysamples,eachconsistingof250checks,wereselectedandexamined.Thenumberofdefectivechecksfoundineachsampleisshownbelow.Example:AttributeControlChExample:AttributeControlChart

Themanagerofthecheckencodingdepartmentknowsfrompastexperiencethatwhentheencodingprocessisincontrol,anaverageof1.6%oftheencodedchecksaredefective. Shewantstoconstructapchartwith3-standarddeviationcontrollimits.Example:AttributeControlChExample:AttributeControlChartExample:AttributeControlChExample:AttributeControlChartExample:AttributeControlChInspectionoftheunitsinthesampleisperformedonavariablebasis.Theinformationprovidedfrominspectingasampleofsizenis:Samplemean,x,orthesumofmeasurementofeachunitinthesampledividedbynRange,R,ofmeasurementswithinthesample,orthehighestmeasurementinthesampleminusthelowestmeasurementinthesampleControlChartsforVariablesInspectionoftheunitsintheInthiscasetwoseparatecontrolchartsareusedtomonitortwodifferentaspectsoftheprocess’soutput:CentraltendencyVariabilityCentraltendencyoftheoutputismonitoredusingthex-chart.VariabilityoftheoutputismonitoredusingtheR-chart.ControlChartsforVariablesInthiscasetwoseparatecontx-ChartThecentrallineisx,thesumofanumberofsamplemeanscollectedwhiletheprocesswasconsideredtobe“incontrol”dividedbythenumberofsamples.The3slowercontrollimitisx-ARThe3suppercontrollimitisx+ARFactorAisbasedonsamplesize.===x-ChartThecentrallineisx,R-ChartThecentrallineisR,thesumofanumberofsamplerangescollectedwhiletheprocesswasconsideredtobe“incontrol”dividedbythenumberofsamples.The3slowercontrollimitisD1R.The3suppercontrollimitisD2R.FactorsD1andD2arebasedonsamplesize.R-ChartThecentrallineisR,3sControlChartFactorsforVariables

ControlLimitFactor ControlLimitFactor Sample forSampleMean forSampleRange Sizen A D1 D2 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.116 10 0.308 0.223 1.777 15 0.223 0.348 1.652 20 0.180 0.414 1.586 25 0.153 0.459 1.5